Angelescu polynomials

Properties

Addition and recurrence relations

The Angelescu polynomials satisfy the following addition theorem:

(-1)^n\sum_{r=0}^m\frac{L_{m+n-r}^{(n)}(x)\pi_r(y)}{(n+m-r)!r!} = \sum_{r=0}^m (-1)^r\binom{-n-1}{r} \frac{\pi_{n-r}(x+y)}{(m-r)!},where L^{(n)}_{m+n-r} is a generalized Laguerre polynomial.

A particularly notable special case of this is when n=0, in which case the formula simplifies to\frac{\pi_m(x+y)}{m!} = \sum_{r=0}^m \frac{L_{m-r}(x)\pi_r(y)}{(m-r)!r!} - \sum_{r=0}^{m-1} \frac{L_{m-r-1}(x)\pi_r(y)}{(m-r-1)!r!}.

The polynomials also satisfy the recurrence relation

\pi_s(x) = \sum_{r=0}^n (-1)^{n+r}\binom{n}{r}\frac{s!}{(n+s-r)!}\frac{d^n}{dx^n}[\pi_{n+s-r}(x)],

which simplifies when n=0 to \pi'_{s+1}(x) = (s+1)[\pi'_s(x) - \pi_s(x)]. This can be generalized to the following:

-\sum_{r=0}^s \frac{1}{(m+n-r-1)!}L^{(m+n-1)}{m+n-r-1}(x)\frac{\pi{r-s}(y)}{(s-r)!} = \frac{1}{(m+n+s)!}\frac{d^{m+n}}{dx^m dy^n}\pi_{m+n+s}(x+y),

a special case of which is the formula \frac{d^{m+n}}{dx^m dy^n}\pi_{m+n}(x+y) = (-1)^{m+n} (m+n)! a_0.

Integrals

The Angelescu polynomials satisfy the following integral formulae:

\begin{align} \int_0^{\infty}\frac{e^{-x/2}}{x}[\pi_n(x) - \pi_n(0)]dx &= \sum_{r=0}^{n-1} (-1)^{n-r+1}\frac{n!}{r!}\pi_r(0)\int_0^{\infty} [\frac{1}{1/2 + p} - 1]^{n-r-1} d[\frac{1}{1/2+p}]\ &= \sum_{r=0}^{n-1} (-1)^{n-r+1}\frac{n!}{r!}\frac{\pi_r(0)}{n-r}[1 + (-1)^{n-r-1}] \end{align}

\int_0^{\infty} e^{-x}[\pi_n(x) - \pi_n(0)]L_m^{(1)}(x)dx = \begin{cases} 0\text{ if }m\geq n\ \frac{n!}{(n-m-1)!}\pi_{n-m-1}(0)\text{ if }0\leq m\leq n-1 \end{cases}

(Here, L_m^{(1)}(x) is a Laguerre polynomial.)

Further generalization

We can define a q-analog of the Angelescu polynomials as \pi_{n, q}(x) := e_q(xq^n) D_q^n[E_q(-x)P_n(x)], where e_q and E_q are the q-exponential functions e_q(x) := \Pi_{n=0}^{\infty} (1 - q^n x)^{-1} = \Sigma_{k=0}^{\infty}\frac{x^k}{[k]!} and E_q(x) := \Pi_{n=0}^{\infty} (1 + q^n x) = \Sigma_{k=0}^{\infty}\frac{q^{\frac{k(k-1)}{2}}x^k}{[k]!}, D_q is the q-derivative, and P_n is a "q-Appell set" (satisfying the property D_q P_n(x) = [n]P_{n-1}(x)).

This q-analog can also be given as a generating function as well:

\sum_{n=0}^{\infty}\frac{\pi_{n, q}(x)t^n}{(1;n)} = \sum_{n=0}^{\infty}\frac{(-1)^n q^{\frac{n(n-1)}{2}}t^n P_n(x)}{(1;n)[1-t]_{n+1}},where we employ the notation (a;k) := (1 - q^a)\dots (1 - q^{a+k-1}) and [a+b]n = \sum{k=0}^n\begin{bmatrix}n\k\end{bmatrix}a^{n-k}b^k.

References